Linear Stability Analysis to Determine Boundary Between Stable or Unstable Flow in Nonlinear Simulation of Vorticity Equation

Abstract

The nonlinear simulation of the vorticity equation is exciting because it provides a more realistic view, it is consistent with the previous investment. However, a linear stability analysis is needed to determine the stability limit of the system, which serves as a guide in determining the flow limits for stable and unstable flows. We use numerical methods to solve this linear stability analysis. This analysis utilizes the Fourier series in the linear form of the vorticity equation, then the vorticity-stream equation containing the Fourier coefficient will be formed in the matrix. Next, numerically using Matlab, we look for the eigenvalues of the matrix to determine the k, beta and Reynolds number values. If one of the eigenvalues has a real positive form given k, beta, and Reynolds number, then the equilibrium solution of the vortex equation is unstable. From graphing the real part of the eigenvalues, which is a function of the Reynolds number for some beta with k = 0.5, we get the critical point of the Reynolds number. By using a Reynolds number greater than the critical Reynolds number give result in an unstable fluid from the simulation. Vice versa, simulation using the Reynolds number, which is smaller than the critical Reynolds number will produce a stable flow. And from the linear stability curve, we know that for k > 1, the flow is stable for every Reynolds number.